Excursions in Modern Mathematics, 7th Edition

Description

Excursions in Modern Mathematics, Seventh Edition, shows readers that math is a lively, interesting, useful, and surprisingly rich subject. With a new chapter on financial math and an improved supplements package, this book helps students appreciate that math is more than just a set of classroom theories: math can enrich the life of any one who appreciates and knows how to use it.

Table of Contents

Part 1. The Mathematics of Social Choice

1. The Mathematics of Voting: The Paradox of Democracy

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)

1.5 The Method of Piecewise Comparisons

1.6 Rankings

Profile: Kenneth J. Arrow

Key Concepts

Exercises

Projects and Papers

References and Further Readings

2. The Mathematics of Power: Weighted Voting

2.1 An Introduction to Weighted Voting

2.2 The Banzhaf Power Index

2.3 Applications of the Banzhaf Power Index

2.4 The Shapely-Shubik Power Index

2.5 Applications of the Shapely-Shubik Power Index

Profile: Lloyd S. Shapely

Key Concepts

Exercises

Projects and Papers

References and Further Readings

3. The Mathematics of Sharing: Fair-Division Games

3.1 Fair-Division Games

3.2 Two Players: The Divider-Chooser Method

3.3 The Lone-Divider Method

3.4 The Lone-Chooser Method

3.5 The Last-Diminisher Method

3.6 The Method of Sealed Bids

3.7 The Method of Markers

Profile: Hugo Steinhaus

Key Concepts

Exercises

Projects and Papers

References and Further Readings

4. The Mathematics of Apportionment: Making the Rounds

4.1 Apportionment Problems

4.2 Hamilton's Method and the Quota Rule

4.3 The Alabama and Other Paradoxes

4.4 Jefferson's Method

4.5 Adams's Method

4.6 Webster's Method

Historical Note: A Brief History of Apportionment in the United States

Key Concepts

Exercises

Projects and Papers

References and Further Readings

Mini-Excursion 1: Apportionment Today

Part 2. Management Science

5. The Mathematics of Getting Around: Euler Paths and Circuits

5.1 Euler Circuit Problems

5.2 What is a Graph?

5.3 Graph Concepts and Terminology

5.4 Graph Models

5.5 Euler's Theorems

5.6 Fleury's Algorithm

5.7 Eulerizing Graphs

Profile: Leonard Euler

Key Concepts

Exercises

Projects and Papers

References and Further Readings

6. The Mathematics of Touring: The Traveling Salesman Problem

6.1 Hamilton Circuits and Hamilton Paths

6.2 Complete Graphs

6.3 Traveling Salesman Problems

6.4 Simple Strategies for Solving TSPs

6.5 The Brute-Force and Nearest-Neighbor Algorithms

6.6 Approximate Algorithms

6.7 The Repetitive Nearest-Neighbor Algorithm

6.8 The Cheapest Link Algorithm

Profile: Sir William Rowan Hamilton

Key Concepts

Exercises

Projects and Papers

References and Further Readings

7. The Mathematics of Networks: The Cost of Being Connected

7.1 Trees

7.2 Spanning Trees

7.3 Kruskal's Algorithm

7.4 The Shortest Network Connecting Three Points

7.5 Shortest Networks for Four or More Points

Profile: Evangelista Torricelli

Key Concepts

Exercises

Projects and Papers

References and Further Readings

8. The Mathematics of Scheduling: Chasing the Critical Path

8.1 The Basic Elements of Scheduling

8.2 Directed Graphs (Digraphs)

8.3 Scheduling with Priority Lists

8.4 The Decreasing-Time Algorithm

8.5 Critical Paths

8.6 The Critical-Path Algorithm

8.7 Scheduling with Independent Tasks

Profile: Ronald L. Graham

Key Concepts

Exercises

Projects and Papers

References and Further Readings

Mini-Excursion 2: A Touch of Color

Part 3. Growth And Symmetry

9. The Mathematics of Spiral Growth: Fibonacci Numbers and the Golden Ratio

9.1 Fibonacci's Rabbits

9.2 Fibonacci Numbers

9.3 The Golden Ratio

9.4 Gnomons

9.5 Spiral Growth in Nature

Profile: Leonardo Fibonacci

Key Concepts

Exercises

Projects and Papers

References and Further Readings

10. The Mathematics of Money: Spending it, Saving It, and Growing It

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings for the Future

Key Concepts

Exercises

Projects and Papers

References and Further Readings

11. The Mathematics of Symmetry: Beyond Reflection

11.1 Rigid Motions

11.2 Reflections

11.3 Rotations

11.4 Translations

11.5 Glide Reflections

11.6 Symmetry as a Rigid Motion

11.7 Patterns

Profile: Sir Roger Penrose

Key Concepts

Exercises

Projects and Papers

References and Further Readings

12. The Geometry of Fractal Shapes: Naturally Irregular

12.1 The Koch Snowflake

12.2 The Sierpinski Gasket

12.3 The Chaos Game

12.4 The Twisted Sierpinski Gasket

12.5 The Mandelbrot Set

Profile: Benoit Mandelbrot

Key Concepts

Exercises

Projects and Papers

References and Further Readings

Mini-Excursion 3: The Mathematics of Population Growth: There is Strength in Numbers